matita/matita/contribs/lambdadelta/ground/relocation/pr_nat_nat.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "ground/arith/nat_lt_pred.ma".
  16 include "ground/relocation/pr_nat.ma".
  17
  18 (* NON-NEGATIVE APPLICATION FOR PARTIAL RELOCATION MAPS *****************************)
  19
  20 (* Main destructions ********************************************************)
  21
  22 theorem pr_nat_monotonic (k2) (l2) (f):
  23         @↑❪l2,f❫ ≘ k2 → ∀k1,l1. @↑❪l1,f❫ ≘ k1 → l1 < l2 → k1 < k2.
  24 #k2 @(nat_ind_succ … k2) -k2
  25 [ #l2 #f #H2f elim (pr_nat_inv_zero_dx … H2f) -H2f //
  26   #g #H21 #_ #k1 #l1 #_ #Hi destruct
  27   elim (nlt_inv_zero_dx … Hi)
  28 | #k2 #IH #l2 #f #H2f #k1 @(nat_ind_succ … k1) -k1 //
  29   #k1 #_ #l1 #H1f #Hl elim (nlt_inv_gen … Hl)
  30   #_ #Hl2 elim (pr_nat_inv_succ_bi … H2f (↓l2)) -H2f [1,3: * |*: // ]
  31   #g #H2g #H
  32   [ elim (pr_nat_inv_push_succ … H1f … H) -f
  33     /4 width=8 by nlt_inv_succ_bi, nlt_succ_bi/
  34   | /4 width=8 by pr_nat_inv_next_succ, nlt_succ_bi/
  35   ]
  36 ]
  37 qed-.
  38
  39 theorem pr_nat_inv_monotonic (k1) (l1) (f):
  40         @↑❪l1,f❫ ≘ k1 → ∀k2,l2. @↑❪l2,f❫ ≘ k2 → k1 < k2 → l1 < l2.
  41 #k1 @(nat_ind_succ … k1) -k1
  42 [ #l1 #f #H1f elim (pr_nat_inv_zero_dx … H1f) -H1f //
  43   #g * -l1 #H #k2 #l2 #H2f #Hk
  44   lapply (nlt_des_gen … Hk) -Hk #H22
  45   elim (pr_nat_inv_push_succ … H2f … (↓k2) H) -f //
  46 | #k1 #IH #l1 @(nat_ind_succ … l1) -l1
  47   [ #f #H1f elim (pr_nat_inv_zero_succ … H1f) -H1f [ |*: // ]
  48     #g #H1g #H #k2 #l2 #H2f #Hj elim (nlt_inv_succ_sn … Hj) -Hj
  49     /3 width=7 by pr_nat_inv_next_succ/
  50   | #l1 #_ #f #H1f #k2 #l2 #H2f #Hj elim (nlt_inv_succ_sn … Hj) -Hj
  51     #Hj #H22 elim (pr_nat_inv_succ_bi … H1f) -H1f [1,4: * |*: // ]
  52     #g #Hg #H
  53     [ elim (pr_nat_inv_push_succ … H2f … (↓k2) H) -f
  54       /3 width=7 by nlt_succ_bi/
  55     | /3 width=7 by pr_nat_inv_next_succ/
  56     ]
  57   ]
  58 ]
  59 qed-.
  60
  61 theorem pr_nat_mono (f) (l) (l1) (l2):
  62         @↑❪l,f❫ ≘ l1 → @↑❪l,f❫ ≘ l2 → l2 = l1.
  63 #f #l #l1 #l2 #H1 #H2 elim (nat_split_lt_eq_gt l2 l1) //
  64 #Hi elim (nlt_ge_false l l)
  65 /2 width=6 by pr_nat_inv_monotonic/
  66 qed-.
  67
  68 theorem pr_nat_inj (f) (l1) (l2) (l):
  69         @↑❪l1,f❫ ≘ l → @↑❪l2,f❫ ≘ l → l1 = l2.
  70 #f #l1 #l2 #l #H1 #H2 elim (nat_split_lt_eq_gt l2 l1) //
  71 #Hi elim (nlt_ge_false l l)
  72 /2 width=6 by pr_nat_monotonic/
  73 qed-.